Question: Puma Grove, the world's best female golfer, is two shots away from winning another tournament. Puma's caddy stands on a marker exactly $200$ meters from the pin and determines the angle between the pin and Puma's ball is $110^\circ$. Standing close to her ball, Puma measures the angle between her caddy and the pin. After performing a calculation, she lines up her shot and swings. Her shot travels $234$ meters directly at the pin (exactly as planned) and lands perfectly next to the hole. What was the measure of the angle Puma saw between her caddy and the pin? Do not round during your calculations. Round your final answer to the nearest degree.
Answer: Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $\angle B=\theta$. $\,\,\,\,234\text{ m}$ $200\text{ m}\,\,\,\,\,$ $A$ $B$ $C$ $\theta$ $110^\circ$ Since we are given two side lengths, we can use the law of sines. When using the law of sines we have to keep in mind the ambiguous case, where the angle can be either acute or obtuse. In our case, as the triangle already has an angle of $110^\circ$, we know $\theta$ must be acute so the ambiguous case doesn't apply. Using the law of sines $\begin{aligned} \dfrac{\sin(B)}{AC}&=\dfrac{\sin(C)}{AB}\\\\ \dfrac{\sin(\theta)}{200} &= \dfrac{\sin(110^\circ)}{234} \gray{\text{Substitute}} \\\\ \sin(\theta) &= \dfrac{200\cdot \sin(110^\circ) }{234} \\\\ \theta &= \sin^{-1}\left(\dfrac{200\cdot \sin(110^\circ) }{234}\right) \\\\ \theta &\approx 53^\circ \end{aligned}$ The answer The measure of the angle Puma saw between her caddy and the pin is $53^\circ$.